1. Ch 5: Equal probability cluster samples
4/19/2017
Cluster sampling
DEFN: A cluster is a group of observation units (or “elements”)
Stat 804

2. Cluster sample
DEFN: A cluster sample is a probability sample in which a sampling unit is a cluster

3. Cluster sample – 2 1-stage cluster sampling
Divide the population (of N elements) into NI clusters (of size Ni for cluster i)
Cluster = group of elements
An element belongs to 1 and only 1 cluster
Sampling unit
Cluster = group of elements = PSU = primary sampling unit
Can use any design to select clusters (ST, PPS)
Data collection
Collect information on ALL elements in the cluster

4. 1-stage CS ST Sample of 40 elements A block of cells is a cluster
A block of cells is a stratum
SU is a cluster
Don’t sample from every cluster
SU is an element (or OU)
Sample from every stratum

5. Cluster vs. stratified sampling
Cluster sample
Divide N elements into NI clusters
Cluster or PSU i has Ni elements
Take a sample of nI clusters
Stratified sampling
N elements divided into H strata
An element belongs to 1 and only 1 stratum
Take a sample of n elements, consisting of nh elements from stratum h for each of the H strata

6. Cluster sample – 3 2-stage cluster sampling Process
Select PSUs (stage 1)
Select elements within each sampled PSU (stage 2)
First stage sampling unit is a …
PSU = primary sampling unit = cluster
Second stage sampling unit is a …
SSU = secondary sampling unit = element = OU
Only collect data on the SSUs that were sampled from the cluster

7. 1-stage vs. 2-stage cluster sampling
1-stage cluster sample (stop here) OR
Stage 1 of 2-stage cluster sample (select PSUs)
Stage 2 of 2-stage cluster sample (select SSUs w/in PSUs)

8. Why use cluster sampling?
May not have a list of OUs for a frame, but a list of clusters may be available
List of Lincoln phone numbers (= group of residents) is available, but a list of Lincoln residents is not available
List of all NE primary and secondary schools (= group of students) is available, but a list of all students in NE schools is not available
May be cheaper to conduct the study if OUs are clustered
Occurs when cost of data collection increases with distance between elements
Household surveys using in-person interviews (household = cluster of people)
Field data collection (plot = cluster of plants, or animals)

9. Defining clusters due to frame limitations
A cluster (or PSU) is a group of elements corresponding to a record (row) in the frame
Example
Population = employees in McDonald’s franchises
Element = employee
Frame = list of McDonald’s stores
PSU = store = cluster of employees

10. Defining clusters to reduce travel costs
A cluster (or PSU) is a group of nearby elements
Example
Population = all farms
Element = farm
Frame = list of sections (1 mi x 1 mi areas) in rural area
PSU = section = cluster of farms

11. Cluster samples usually lead to less precise estimates
Elements within clusters tend to be correlated due to exposure to similar conditions
Members of a household
Employees in a business
Plants or soil within a field plot
We are getting less information than if selected same number of unrelated elements
Select sample of city blocks (clusters of households)
Ask each household:
Should city upgrade storm sewer system?
PSU (city block) 1
No storm sewer  households will tend to say yes
PSU (city block) 2
New development  households will tend to say no

12. Defining clusters for improved precision
Define clusters for which within-cluster variation is high (rarely possible)
Make each cluster as heterogeneous as possible
Like making each cluster a mini-population that reflects variation in population
Minimizes the amount of correlation among elements in the cluster
Opposite of the approach to stratification
Large variation among strata, homogeneous within strata
Define clusters that are relatively small
Extreme case is cluster = element
Decreasing the number of correlated observations in the sample

13. Example for single-stage cluster sampling w/ equal prob (CSE1)
Dorm has NI = 100 suites (clusters)
Each suite has Ni = 4 students (4 elements in cluster i , i = 1, 2, … , NI)
Note that there are
Take SRS nI = 5 suites (clusters)
Ask each student living in each of the 5 suites
How many nights per week do you eat dinner in the dining hall?
Will get observations from a sample of 20 students = 5 suites x 4 students/suite

14. Dorm example – 2 Stu-dent Suite 6 Suite 21 Suite 28 Suite 54 Suite 893 6 2 4
Total 20 14 19 21 10

15. Dorm example – 3 SRS of nI = 5 dorm rooms
Data on each cluster (all students in dorm room)
ti = total number of dining hall dinners for dorm room i
t2 = 14 dining hall dinners for 4 students in dorm room 2
Estimated total number of dining hall nights for the dorm students
HT estimator of total = pop size x sample mean (of cluster totals)

16. Notation Response variable for SSU j in PSU i yij
e.g., age of j-th resident in household i
e.g., whether or not dorm resident j in room i owns a computer

17. Cluster-level population parameters (for cluster i )
Cluster size =
Cluster population total
Note that we observe cluster population total (or mean or variance) for each sample cluster in 1-stage cluster sampling
We will estimate cluster parameters in 2-stage cluster sampling
Ni elements

18. Popuation
1-stage cluster sample

19. Data from cluster samples
Work with element and cluster-level data
Element data set will have columns for
Cluster id
Element id within cluster
Variable (y)
Will also summarize this data set to generate cluster parameters (1-stage) or estimates of cluster parameters (2-stage)
Cluster total (or estimate)
Cluster mean (or estimate)
Cluster variance (or estimate)

20. 1-stage cluster sample Element data Cluster summaryi j
yij 1
y11 2
y12 3
Y13 4
y14
y21
y22
y23
y31 … i ti 1 t1 2 t2 3 t3 …

21. CSE1 unbiased estimation under SI – total t
Estimator for population total using data collected from a 1-stage cluster sample
SI of clusters
Estimator of variance of

22. Dorm example – 4
Estimated population total
Estimated variance

23. Dorm example – 5 Inclusion probability for student j in dorm room i
N = 100 dorm rooms
n = 5 sample dorm rooms
Take all 4 students in dorm room
ij = nI / NI = 1/20 = 0.05